課程名稱 |
代數導論一 Introduction to Algebra (Ⅰ) |
開課學期 |
101-1 |
授課對象 |
電機資訊學院 電機工程學系 |
授課教師 |
王姿月 |
課號 |
MATH2105 |
課程識別碼 |
201 24210 |
班次 |
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學分 |
3 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期一3,4(10:20~12:10)星期四7,8(14:20~16:20) |
上課地點 |
新204新204 |
備註 |
總人數上限:100人 外系人數限制:10人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1011abstactalgebraW |
課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
課程大綱
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課程概述 |
*Contents:
1. Integers and Permutations: divisors and prime factorization, integers modulo n, permutations
2. Groups: groups, subgroups, cyclic groups, homomorphisms, cosets, Lagrange theorem, groups of motions and symmetries, isomorphism theorem
3. Rings: examples and basic properties, integral domains and fields, ideal and factor rings, homomorphisms, polynomial rings, symmetric polynomials, unique factorization domains, principal ideal domain
4. Fields: algebraic extensions, splitting fields, finite fields, geometric construction(ruler and compass), fundamental theorem of algebra
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課程目標 |
Understand the structure of groups, rings, and fields. |
課程要求 |
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預期每週課後學習時數 |
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Office Hours |
每週五 15:00~16:00 每週一 15:00~16:00 |
指定閱讀 |
教科書: W. K. Nicholson, Introduction to Abstract Algebra, 3rd edition, John Wiley & Sons, 2007 |
參考書目 |
T. W. Judson, Abstract Algebra Theory and Applications
(can be obtained via http://abstract.pugetsound.edu/download.html
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評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
作業 |
20% |
Homework will be collected every Thursday (tutorial class). No Late Homework! |
2. |
平時考 |
20% |
(Thursday tutorial class) 20 minutes for each test, total of 7-10 tests for this semester |
3. |
期中考 |
30% |
the Week of November 4 |
4. |
期末考 |
30% |
the Week of January 7 |
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週次 |
日期 |
單元主題 |
第1週 |
9/10,9/13 |
* Definition and example of groups, rings, and fields
* Division Algorithm, Euclidean Algorithm
*equivalence relation
*Integers modulo n
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第2週 |
9/17,9/20 |
Chinese remainder theorem, Fermat's theorem, an application to crytography, permutations, binary operations |
第3週 |
9/24,9/27 |
definition of groups, basic properties of groups, classification of groups with 2,3,or 4 elements,
subgroups |
第4週 |
10/01,10/04 |
examples of subgroups, cyclic groups, order of an element, fundamental theorem of finite cyclic groups, examples of group homomorphism and group isomorphism |
第5週 |
10/08,10/11 |
basic properties of homomorphism and isomorphism, automorphism, inner automorphism, Cayley's theorem, cosets, Langrange's theorem |
第6週 |
10/15,10/18 |
Application of Lagrange's theorem, Dihedral groups, classification of groups of order 6, Euler's theorem, normal subgroups |
第7週 |
10/22,10/25 |
classification of order 8 abelian groups, definition of factor groups (quotient groups) and basic properties, commutator subgroups (derived groups), alternation groups (A_n) |
第8週 |
10/29,11/01 |
kernel, image, isomorphism theorem, homomorphic images of a group, alternating groups (A_n) |
第9週 |
11/05,11/08 |
simple groups, examples of simple groups, more on alternating groups, review for midterm exam |
第10週 |
11/12,11/15 |
definition of rings, examples, basic properties, characteristic of a ring, subrings, units, idempotents, nilpotents, division rings, fields |
第11週 |
11/19,11/22 |
integral domains, fields, field of quotients, ideals, factor rings |
第12週 |
11/26,11/29 |
prime ideals, maximal ideals, ring homomorphisms, isomorphism theorem |
第13週 |
12/03,12/06 |
applications of the isomorphism theorem, decomposition of rings, Chinese remainder theorem, polynomial rings, division algorithm of division ring, evaluation theorem, remainder theorem |
第14週 |
12/10,12/13 |
factorization of polynomial rings over fields, Gauss lemma, Eisenstein criterion |
第15週 |
12/17,12/20 |
factor rings of polynomials over a field, Kronecker's theorem, algebraic extension of a field, minimal polynomial of an algebraic element over a field |
第16週 |
12/24,12/27 |
splitting fields, finite fields |
第17週 |
12/31,1/03 |
geometric construction(ruler and compass), fundamental theorem of algebra |
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